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# Quantum mechanics

Schrodinger equation   | Matrix mechanics   |   Dirac Equation   |   Quantum electrodynamics

# Theory of relativity

Special theory of relativity   |   General theory of relativity

# Matrix mechanics pdf

This is most modern and precise theory of atomic phenomena. Matrix mechanics reveals that the occurances that happen inside atom are probalistic. We have to work with a collection of data instead of certain changes governed by cause and effect relationship. The position observable is a matrix which has an infinite number of elements each of which is a possible measureable quantity. Momentum observable is also a matrix. What we can best expect is that certain change is more likely than the other. Only statistical knowledge about the atom can help us determine which change is more likely than other. Thus a new kind of mathematics has been developed , which compels us to describe everything about the atom in terms of probability. This kind of uncertainty is inherent in the quantum reality. Heisenberg picture of atom is related to operators A and the Hamiltonian H which is itself an operator. The time evolution of such operator A is the basic equations of motion according to Heisenberg;

# Eigenvector

Eigenvector is a vector which produces the same vector multiplied by some constant when it is acted by some linear operator like a matrix A. In quantum mechanics the operator H (hamiltonian) acts on wave function eigenvector and produces a corresponding eigenvalue E.

Heigenberg's theory of atom treats subatomic particles differently than ordinary quantum theory. According to this theory atom or electron has no immediate reality as the objects of sense but only those properties ascribed to light wave. The troubles came , from trying to picture the atom or electron as in ordinary space. These entities must not be said to be in definite position at some time. If we want to retain the corpuscular model of electron , we can do it not assigning definite position to an electron. it must be replaced with a group of physical entities which represent the place of the electron. These quantities can be called physically observable quantities or radiations in the form of light.

"There is a difference between knowing the path and walking the path.."

# Heisenberg's Uncertainty relation

The atomic model of Bohr had limitation due to its hypothetical orbits which can not be observed. Heisenberg thought this as a nonphysical description of the atom. So Heigenberg made a reasoning that the radiation that the electron emits can be taken as a physical observable. According to Bohr's theory an electron jumps from one orbit to another when it emits or absorbs light or electromagnetic energy without being in the intermediate space. The frequency of emitted radiation from an atom can be represented by two variable m and n , which represent two Bohr's orbits respectively. Comparing it with classical harmonic motion , which can be represented by Fourier series, Heisenberg showed that two observables x(t), p(t) can be correlated with this Fourier series. Each of these observables is a matrix (row m and column n). Thus atom can be completely modeled by two matrices : position and momentum matrices (with m rows and n columns) with frequency of radiation determined by plank's formula.

The Fourier series representation of harmonic motion is :

The equation of motion reasoned by Heigenberg has the following Fourier components which itself must oscillate at frequency W(mm) :

Observable quantities like phase, frequency and amplitude of radiation thus enter into the mathematical equation. But as the those are the quantities that depends on difference of two number, the position and momentum will be matrices of m rows and n columns. Sum of all the matrix elements thus has no physical significance as it represents just a wave.

# Momentum and position matrix P(0), X(0)

At certain moment (t) the position and momentum has to take values as matrices:

As it is evident that two matrices does not always commute , there is no well defined position and momentum of any particle at the same instant. The order of product of two matrices changes the value of multiplication. If two matrices each have the same identical elements then the order of their product do not change the value of multiplication.

In case with the multiplication of vector with a matrix the resultant value of the multiplication is also a vector:

If each contains distinct elements , the order of multiplication changes the value of multiplication. As position and momentum, in general , do not contain identical elements , their product depend s on their order of multiplication. Such was the reasoning of Heisenberg. No state possesses definite position and momentum simultaneously. The next is general result of such anti-commutation. The result of the difference is number i (imaginary) multiplied by plank constant.

The product of variance of position and momentum , σ(x) and σ(p) can be proven to be less than h/2. This is the uncertainty that saves quantum mechanics from breaking down. A proof using standard deviation of operators can be made simply:

Any operator O that acts on wave wave function ψ must have standard deviation σ

More compact and popular version of the uncertainty principle is :

There is some misconception about the uncertainty principle. This is not related to limitation of the precision of our observing instruments. Now matter how precise our measuring intruments are , there will be always some uncertainty or unpredictability. The uncertainty principle is intrinsic property of quantum world. This is the way our world in the smallest scale behaves. Due to this uncertainty principle particle and anti -particle always appear and annihilate in vacuum out of nothing. Classical objects like billiard balls, moon have definite position and momentum. We can determine their position and momentum simultaneouly. But when we try to measure the position of tiny objects like electron the very act of observation changes the momentum of the particle. Some momentum of the incident light is transferred to the electron. This change is random. As a result, we can not predict with certainty the exact momentum of the electron at the same time we measure its position.

## Free Schrodinger equation

Schrodinger equation plays a vital role in quantum mechanics. We set potential energy equal to zero in free Shcrodinger equation. The solution of such equation will then change identically.

## Heisenberg's equation of motion

Heiseberg's equation of motion describes the time evolution of operator A in heisenberg's picture. In heisenberg's picture operators carry time dependence while the state vector remains constant. The proof of equation's of motion is as follows:

The time derivative of operator O(s) is zero which is omitted in the first line. Terms:
O(s) = operator is Schroinger picture
O(h) = Operator in Heisenberg's picture.
U = unitary operator desrcibing time evolution.

## Some useful equations in quantum mechanics

Erhenfest's theorem :

View this video for more explanation:

expectation value of a random experiment is the average value of all the outcomes of the experiment. Suppose you throw a six headed dice one time. Then the expectation value will be (1+2+3+4+5+6)/6 = 3.5. Similarly operators in quantum mechanics have expectation value. Operators act on observables to give outcome of quantum measurements. So it is natural that they have expectation values.

"there are some corners of the universe which have breed the most terrible things. Things that work against everything we believe in. These things must be fought "

### Weirdness of quantum world

Quantum world is bizzare and mysterious. As for the atom, it gives us no clues of its existence untill it gives off radiation into outer world. The radiations come into discrete chunks of energy. That was the theory of Max Planck. If Max Planck did not invent his law we would be burned by the sun's ultraviolate rays. The smallness of planck's constant h makes the quantum world behaves differently than classical world.

### Quantum cosmology

Vector, scalar and tensor harmonics on three sphere are introduced in order to study gravitational physics. These harmonics are the eigenfunctions of covariant laplace operator which satisfy some divergence and trace identity ,and ortho-normality conditions.

The wave function is computed using these harmonics and corresponding Schrodinger equation can be formulated easily.

De sitter metric is a solution of Einstein's field equations. It was found by Astronomer De sitter. It is given as a static form which is like Swardchild's solution in case of non-rotating massive objects.

like Swardchild solution de sitter metric contains a singularity at some r -value. The n-dimensional de Sitter space, denoted by dS(n), is the Lorenzian manifold which is the analog of n-sphere(with its Riemanian metric), and which has positive constant curvature and simply connected for at least n=3.

Quantum mechanics have been merged with special relativity successfully. This apparent marriage has brought out some difficulties. Qantum mechanics is regarded as a complete and most succesful theory. But this theory is not free of contradiction. It has resulted in a paradoxical phenomena called measurement paradox. Measuring a quantum system is depenent on the system measuring it. Observing a particular quantum phenomena does not imitate classical behavior. Prior to measurement the system posesses a myriad possibilities to be evident when measured. This is so called measurement paradox.

On the other hand when special relativity is combined with quantum mechanics , infinities creep into the calculution. It is not desirable to work with infinties in real world situation. In general theory of relativity singularities arise because of very dense mass and energy distribution. Laws of physics breaks down and can not predict anything in such a case.

wheeler Dr with equation is a functional differential equation to combine features of quantum mechanics with General relativity. A functional differential equation is a equation which relates a derivative of a functional to the change of the function on which the functional depends.

Einstein said that "God does not play dice" as he was an opponent of quantum mechanics. Stephen Hawking claimed that God not only play dice but He throws them where we can not see them. Here is a visualization of black hole

from where nothing can escape. Everything else which enters a black hole's event horizon becomes lost inside the singularity.

## Bohr's correspondence principle

Bohr's correspondence principle states that in some limit when the quantum number n is very large the quantum description and classical description give the same result. It is the cosequences of Bohr's own theory. An electron , when quantum number is very large, can be regarded as free. Then it behaves classically. Mathematical formalism is

In terms of wavefunction ψ this can also be defined or reinterpreted.

How is the classical probablity derived? Foods for thoughts.
The same arguments go for angular momentum of electron

## Compton effect

Compton effect is the scattering of a x-ray particle by electron. The momentum of the x-ray particle knocks off the electron from its position and changes its direction by some small angle. Here is the graphical representation and vector analysis.

## Common sense has no place in quantum mechanics..

Electronic band theory is entirely based on quantum mechanics. Electronic band theory explains the electrical properties of various materials like insulators, conductors and semi-conductors. It is explained in Integrated electronics page. There is a close connection between inter-molecular distance and band gaps between energy levels.

Sometimes atomic spectrum is explained by multiple fourier series. Let us take a f(x,y) is a continuously differentiable function with a periodic of 2π in both of the variables. That is f(x, y) = f(x+2π , y+2π ). Then the fourier expansion of the given function will be

## Atomic spectroscopy

Atomic spectroscopy is a method of analyzing electromagnetic radiation emitted or absorbed by the atoms. In the sun due to nuclear fusion , radiation is emitted from atoms of hydrogen. The radiation reaches the earth and we get heat energy. By analysing the radiation the nature of atom and its properties can be determined. So spectroscopy is very important in studying the composition of our sun and other planets. It is very modern approach.

After the development of matrix mechanics quantum mechanics were understood completely. The next revolution was to come from Dirac's formulation of relativistic quantum mechanics. He successfully merged quantum mecahnics with Einstein's theory of special relativity. Dirac by analyzing his equation predicted anti-matter should exist. Hydrogen atom can be described by rlativistic quantum mechanics.

Before proceding further it is better to recapitulate some of the equations of non-relativistic quantum mechanics.

## Secret of creation can only be traced through quantum mechanics..

Quantum theory of paramagnetism is well-established.

The paramagnetism is much related to diamagnetism

## Zero point energy of quantum system

Zero point energy is the lowest energy that a system can have. In quantum harmonic oscillator this energy is shown to be hω/2

We can actually derive this zero-point energy which is also applicable to quantum fields. Near the bottom of the well we have the hamiltonian given by

On the other hand from heisenberg's uncertainty principle

The expectation value of kinetic energy and potential term must satisfy

So the expectation value of the hamiltonian must be at least

where ω = square root k/m. The symmetry of the gauge field can be of two types : one is global and the other is local. Global gauge invariance is applicable to all points in space and time.

global gauge invariance leads to charge conservation law, which is a consequence of Noether theorem.

## Monatomic chain of identical atoms with nearest neighbors

If a large number of atoms are arranged in a row , an atom will attract the other neighboring one. In this way a combined motion will be produced. If all the atoms are in simple harmonic oscillation with the same frequency then the following conditions will hold

The potential function, in this case , can be approximated as

There is a mathematical expression for the thermal dependence of mean energy of a quantum harmonic oscillator.

## Analysis of Hydrogen atom

Schrodinger's equation can be directly applied to Hydrogen Atom

By separation of variable the radial part of the equation can be found as

Schrodinger's equation is an eigenvalue problem which can be solved using the determinant of a matrix

The zeeman effect in hydrogen atom is the splitting of energy of hydrogen orbital

### Reference materials:

Fundamentals of quantum mechanics
A briefer history of time by S. Hawking
A brief history of time by S. Hawking
Quantum mechanics
Grand Design by Stephen Hawking
Higher Engineering Mathematics ( PDFDrive.com ).pdf

## Quantum entanglement

Quantum entanglement is the most bizzare phenomena that is predicted by quantum mechanics.

Quantum phase estimator can be used to measure phase of wave function

## Heisenberg's original paper

Quantum mechanical state of hydrogen atom is characterized by four quantum numbers – principal (n), azimuthal (l), magnetic (m), and spin (s). But energy is dependent on only n, and is independent of the other three quantum numbers. This fact is usually explained by invoking the symmetry present in hydrogen atom. The non-dependency of energy on m is attributed to the spherical symmetry of hydrogen atom, but non-dependency on l is not an obvious symmetry. this non-obvious symmetry is known as 'hidden' or 'dynamical' symmetry
According to classical electrodynamics, orbits of electrons are periodic with periodicity being related to the harmonics of mechanical frequency, i.e., ω, 2ω, etc. As the electron emits electromagnetic radiation, it looses energy. Consequently, radius of orbit becomes smaller, i.e., it exhibits a spiral motion. Although, if the loss of energy is much lesser compared to that of electron’s energy, one can neglect the dissipation and electronic motion can be assumed to be periodic
Classically, with certain approximations, one expects the spectrum for high energy hydrogen atom to be harmonic . This fact can be represented by Fourier expansion of position as follows

a_n is related to the intensity of the corresponding harmonics. In the high quantum number limit, the hydrogen atom do exhibit a harmonic spectrum. For example, if we use Rydberg formula ,

to calculate transition frequencies from n1 = 500 to n2 = 499, 498, 497, 496 , and if n_500-499 = Ω_0 then other frequencies in terms of Ω_0 is 2Ω_0, 3Ω_0 and 4Ω_0 respectively
Heisenberg started with this analogy. He asked the question that if classically, the frequency and intensity of emitted radiations contain information of underlying motion of electron, then why not expect the same from quantum mechanics. Therefore, he thought about the reverse problem and tried to form quantum mechanical position (xqm(t)) and momentum from an(ω), and exp (−inωt). Heisenberg started with two observables in case of hydrogen atom,
1. The transition frequency (n1 to n2).
2. The transition intensities.
It is important to note that quantum mechanics only accounts the observable quantities. For example, as the orbit of the hydrogen atom was not an observable quantity, Heisenberg did not put any attempt to give any physical meaning to it. So, Heisenberg had two pieces of information in his hand – the transition intensities a (n1, n2) and transition frequencies ω(n1, n2). It is apparent that these quantities are 2-index entities as compared to their classical counterparts which are typically 1-index objects. Heisenberg realized that the quantum mechanical position should be mathematically related to these 2-index objects, a(n1, n2) exp (−inω(n1, n2)t). So, he already had the idea that position in quantum mechanics can have a very different meaning, and he was also able to define x_qm mathematically. Then he tried to discover the algebra of this object by exploring the multiplication of two x_qm. To discover that he again resorted to the classical counterpart x^2. Classically

Where

So classically, x(t) 2 is again a Fourier series, whose coefficients are given by (3). It is logical at this point to expect that similar to its classical counterpart, quantum mechanically x_qm(t)^2 should also be a fourier series too .
exp(-ipΩ(m,n)t) .
But the question is how, the new Fourier coefficients (a_m,n) and frequencies (ω_m,n) are related to the Fourier coefficients corresponding to the expansion of x(t)? To explain this relation, Heisenberg again looked towards another experimental observation aka the Ritz combination principle which says
a(m,n) = a(m, n1) + a(n1, n) .
So multiplication of two 2-index object should accord with Ritz principle. Hence the only way the two-index object can be multiple is

and in general

Thus, from simply analyzing x(t)^2, Heisenberg discovered the algebra of these new 2-index objects. Another very important thing that Heisenberg observed was the non-commutability of different quantum mechanical observables. If we take two observables x_qm(t), y_qm(t), then contrary to classical mechanics, their product x_qm(t)y_qm(t) need not be always equal to y_qm(t)x_qm(t). Now, let us put this algebra in a more modern way so that it becomes familiar. The expression x_qm(t) can be arranged in the form of an array,

Now, one can see that if this array is multiplied in Heisenberg’s way, then this array is an example of matrix which can be found in any undergraduate mathematics book. It is obvious now that these 2-index objects represent matrices. We know that matrices do not commute always as Heisenberg noticed. Historical fact is, Heisenberg did not know that he was replacing the classical numbers with matrices. It was Heisenberg’s mentor Born who recognized that these were actually matrices, and within a few months, with the help of his assistant Pascal Jordan and Heisenberg, he was able to present a more robust formalism of quantum mechanics in terms of these new 2-index objects or matrices. This formulation is known as the ‘matrix mechanics method’. As a result Heisenberg made the following comment.
The amplitudes may be treated as complex vectors, each determined by six independent components, and they determine both the polarization and the phase. As the amplitudes are also functions of the two variables n and α, the corresponding part of the radiation is given by the following expressions

## Bremsstrahlung

fast moving charged particle is decelerated in the Coulomb field of atoms. A fraction of its kinetic energy is emitted in form of real photons.

The probability of this process is proportional to 1/M^2 and Z^2 (atomic number of the matter).
Above a few tens MeV, bremsstrahlung is the dominant process for e- and e+ in most materials. It becomes important for muons (and pions) at few hundred GeV
The differential cross section is specified by the Bethe-Heitler formula [Heitl57], corrected and extended for various effects:
• the screening of the field of the nucleus
• the contribution to the brems from the atomic electrons
• the correction to the Born approximation
The polarization of matter.
Screening effect is caused by the electron cloud in the nucleus. The Coulomb field of the nucleus is more or less screened by this electron cloud.
Born approximation is β => αZ . If this condition is violated then the Coulomb wave would be used instead of plane waves.
High energy regime is

Above few GeV the energy spectrum Bethe-Heitler formula becomes simple

It is the required cross section for bremstrahlung.
Mean rate of energy loss due to bremstrahlung is

n_at is the number of atoms per unit volume. The integration immediately gives

The main summary is that the average energy loss per unit path length due to the bremsstrahlung increases linearly with the initial energy of the projectile.

## Defining cross section used in bremstrahlung

The cross section is the effective area that quantifies the likelihood that a given interaction between an incident object and a target object will occur. The term "differential cross section" is more oftern used instead of cross section. The cross section of a particle is the same as that of a hard object if the probabilities of hitting them are the same. It is defined as
σ = μ/n
Where μ is the attenuation coefficient due to the occurence of the event and n is the number of particles per unit volume.

## Hyperfine structure

The interaction between the spin of the nucleus and the angular momentum of the electron causes a further (hyperfine) splitting of atomic states. It is called hyperfine because it is also order α^2 like the fine structure corrections, but it is lesser by a factor of about m_e\m_p because of the mass dependence of the spin magnetic moment for particles.
The magnetic moment of the nucleus is

Where I is the nuclear spin vector. Because the nucleus, the proton, and the neutron have internal structure, the nuclear gyromagnetic ratio is not just 2. For the proton, it is g_p ≈ 5.56. In fact we can calculate the hyperfine contribution to the hamiltonian for l = 0 states

We define the total spin as F = S + I
It is in the states of definite f and m_f that the hyperfine perturbation will be diagonal. In essence, we are doing degenerate perturbation. We could diagonalize the 4 by 4 matrix for the perturbation to solve the problem.

For the hydrogen ground state we are just adding two spin 1/2 particles so the possible values are f=0,1 . The transition between the two states gives rise to EM waves with λ=21 cm.
We will find the effect of an external B field on the Hydrogen hyperfine states both in the strong field and in the weak field approximation. We also work the problem without a field strength approximation. The always applicable transitory field strength result is that the four states have energies which depend on the strength of the B field. Two of the energy eigenstates mix in a way that also depends on B. The four energies are

## If you want to reveal the secret of the universe think in terms of energy , frequency and vibration and if you want to find secret of mathematics think in terms of relations alone...

As explained fourier series plays a vital role in atomic physics. Fourier series can be represented with complex numbers.

A poetry related to particle physics goes like this
Three quarks for Muster Mark! Sure he hasn’t got much of a bark And sure any he has it’s all beside the mark.

## 3D symmetric HO in spherical coordinates

We have already solved the problem of a 3D harmonic oscillator by separation of variables in Cartesian coordinates. It is instructive to solve the same problem in spherical coordinates and compare the results. The potential is

Write the equation in terms of the dimensionless variable

Plugging these into the radial equation, we get

For large y , the behavior will be

Also for small y the behaviour is

Explicitly put in this behavior and use a power series expansion to solve the full equation.

We now need to compute the derivatives

We can now plug these into the radial equation.

Each term will contain the exponential e^{-y^2/2}, so we can factor that out. We can also run a single sum over all the terms.

The terms for large y which go like y^{l+k+2} and some of the terms for small y which go like y^{\ell+k-2} should cancel if we did our job correctly.

Now as usual, the coefficient for each power of y must be zero for this sum to be zero for all y. Before shifting terms, we must examine the first few terms of this sum to learn about conditions on a_0 and a_1. The first term in the sum runs the risk of giving us a power of y which cannot be canceled by the second term if k< 2. For k = 0, there is no problem because the term is zero. For k=1 the term is (2l+2)y^{l-1} which cannot be made zero unless a_1 = 0;
This indicates that all the odd terms in the sum will be zero, as we will see from the recursion relation. Now we will do the usual shift of the first term of the sum so that everything has a y^{l+k} in it.

For large k,

Which will cause the wave function to diverge. We must terminate the series for some k=n_r=0,2,4..., by demanding

These are the same energies as we found in Cartesian coordinates. Lets plug this back into the recursion relation.

To rewrite the series in terms of y^2 and let k take on every integer value, we make the substitutions n_r _> 2n_r and k -> 2k in the recursion relation for \a_{k+1} in terms of \$a_k.

The table shows the quantum numbers for the states of each energy for our separation in spherical coordinates, and for separation in Cartesian coordinates. Remember that there are 2l +1 states with different z components of angular momentum for the spherical coordinate states.

## More quantum field theory formulas

Feynman propagator gives that probability that a particle can go from some point in spacetime to another point. One thing is to remember that in quantum mechanics and quantum field theory everything is described in terms of complex numbers. The Feynman propagator is

The expression involving T is the time-ordered product. The Feynman-propagator can be written as

Relabeling ~p by −~p in the second term gives

The term in brackets can be rewritten as a complex contour integral. As a preparation recall Cauchy’s integral formula.
Photoelectric effect was discovered by Einstein and at the same time Einstein proved that light was consisted of small packet of energy called photons.

The wavefunction of a particle in square potential well is defined as

## De-broglie wave

The possibility that particles of matter like electrons could be both particle-like and wave-like was first proposed by Louis de Broglie in 1923. Specifically he proposed that a particle of matter with momentum p could act as a wave with wavelength
λ = h/p where h is the planck's constant. p is the momentum.
This wavelength is now called the de Broglie wavelength. It is often useful to write the de Broglie wavelength in terms of the energy of the particle. The general relation between the relativistic energy E and the momentum p of a particle of mass m is

This implies that the de Broglie wavelength of a particle with relativistic energy E is given by

When the particle is ultra-relativistic we can neglect mass energy mc^2 and obtain λ = h/ε
When the particle is non-re lativistic, we can set E = mc^2 + E, where E = p^2/2m is the kinetic energy of a non-relativistic particle, and obtain

## Uncertainty relation

We shall introduce the uncertainty principle for the position and momentum of a particle by considering a famous thought experiment due to Werner Heisenberg in which the position of a particle is measured using a microscope. The particle is illuminated and the scattered light is collected by the lens of a microscope as shown in Figure below.
Because of the wave-like properties of light, the microscope has a finite spatial resolving power. This means that the position of the observed particle has an uncertainty given approximately by

where l is the wavelength of the illumination and 2α is the angle subtended by the lens at the particle. We note that the resolution can be improved by reducing the wavelength of the radiation illuminating the particle; visible light waves are better than microwaves, and X-rays are better than visible light waves.

However, because of the particle-like properties of light, the process of observation involves innumerable photon±particle collisions, with the scattered photons entering the lens of the microscope. To enter the lens, a scattered photon with wavelength l and momentum h=l must have a sideways momentum between

Thus the sideways momentum of the scattered photon is uncertain to the degree

The sideways momentum of the observed particle has a similar uncertainty, because momentum is conserved when the photon scatters.
So we get from previous relation
∇x∇p = h
which is approximately the same as Original Heisenberg's uncertainy relation. It asserts that greater accuracy in position is possible only at the expense of greater uncertainty in momentum, and vice versa. The precise statement of the principle is that the fundamental uncertainties in the simultaneous knowledge of the position and momentum of a particle obey the inequality

## Quantum tunnelling

Qunatum tunelling is a quantum phenomena . Through quantum tunnelling a particle can disappear from one side of a potential barrier and can reappear on the other side of it. Using the principle of classical mechanics such phenomena can not be explained. It is only possible in quantum world. Using quantum tunnelling sending some particle faster than the speed of light is possible.
The ability to penetrate and tunnel through a classically forbidden region is one of the most important properties of a quantum particle. To keep the mathematics as simple as possible we shall consider a particle in a simple potential energy field of the form

As shown in Figure below, we have a square barrier of height VB separating the regions ( -&infinity; < x < 0) and (a < x < 1).
We shall see that when a quantum particle encounters the barrier, the outcome is uncertain. Most importantly, we shall show that the particle may be transmitted even when its energy is below VB, and we shall calculate the probability for this to happen The behaviour of a particle of mass m in the potential V(x) is described by a wave function ψ(x, t) which is a solution of the SchroÈdinger equation

To describe the dynamics of the uncertain encounter with the barrier, we seek a wave function C(x, t) which describes an incoming particle and the possibility of reflection and transmission. This wave function should give rise to an incoming pulse of probability representing a particle approaching the barrier

before the encounter. After the encounter, there should be two pulses of probability, one representing the possibility of a reflected particle and the other the possibility of a transmitted particle.
Provided the uncertainty in the energy of the particle is small compared with the variations in the potential energy V(x), we can calculate the probabilities of reflection and transmission by considering a stationary state with definite energy. Such a state is represented by the wave function

where ψ_E(x) is an eigenfunction with energy E satisfying the eigenvalue equation

On the left of the barrier, the potential energy V(x) is zero and the eigenfunction ψ_E(x) satisfies the differential equation

The solution representing an incident wave of intensity |AI|^2 and a reflected wave of intensity |AR|^2 is

The form of the eigenfunction inside the barrier depends on whether the energy of the particle is above or below the barrier. When E > V_B, the region (0 < x < a) is a classically allowed region and the eigenfunction is governed by

The general solution involves two arbitrary constants and it undulates with wave number kB as follows

When E < VB, the region (0 < x < a) is a classically forbidden region. Here, the eigenfunction is governed by

and the general solution is

where B and B` are arbitrary constants. As a result there is an expression for wavefunction on the other side of the barrier. The value of the wavefunction is finite everywhere except at the infinity where r= &infinity;. At that limit the wavefunction vanishes. Due to this finite value everwhere outside the barrier the electron can appear there anytime. This is called quantum tunnelling. It is absurd yet subatomic world is being governed by this phenomena all the time.
Weak nuclear decay happens because there happens quantum tunnelling from nucleus to the outside world of atom. Electron or beta particle overcome the coulomb's potential and come out of the atom.

## Linear superpositions of sinusoidal waves

Two sinusoidal waves moving in opposite directions may be combined to form standing waves. For example, the linear superposition
Acos(kx + Ωt) + Acos(kx - Ωt)
gives rise to the wave 2Acos kx cos Ωt. This wave oscillates with period 2π! and undulates with wavelength 2π/k, but these oscillations and undulations do not propagate; it is a non-Mexican wave which merely stands and waves.
Alternatively, many sinusoidal waves may be combined to form a wave packet. For example, the mathematical form of a wave packet formed by a linear superposition of sinusoidal waves with constant amplitude A and wave numbers in the range k - ∇k to k + ∇k is

If k is positive, this wave packet travels in the positive x direction, and in the negative x direction if k is negative. The initial shape of the wave packet, i.e. the shape at t = 0, may be obtained by evaluating the integral

This gives

A wavepacket is a similar superposition of many waves.

The initial shapes of the wave packets given by a linear superposition of sinusoidal waves with constant amplitude A and wave numbers in the range k - ∇k to k + ∇k . The three diagrams show how the length of a wave packet increases as the range of wave numbers Dk decreases. The value of ADk is constant, but ∇k equals k=8 in diagram (A), ∇k equals k=16 in diagram (B) and ∇k equals k=32 in diagram (C). In general, the length of a wave packet is inversely proportional to Dk and becomes infinite in extent as ∇k ! 0:
A non-dispersive wave has a dispersion relation of the form Ω = ck, where c is a constant so that the velocity of a sinusoidal wave, Ω/k = c,
is independent of the wave number k. A wave packet formed from a linear superposition of such sinusoidal waves travels without change of shape because each sinusoidal component has the same velocity For waves travelling in three dimensions, it has the form

Every wavepacket has a group velocity , It is defined by

Condition for constructive wave interference is

## Quantum mechanics says that reality is not indepent of observers. Everytime we look we create it...

More equations of quantum mechanis are given for general acquaintance.

## Bound states

If a bound state exists, it has a negative energy somewhere between E = -V_0 and E = 0. We shall set E = - ε, where E is the binding energy, and seek solutions of Schrodinger equation. In the region ( &infinity; < x < 0), the potential energy is infinite and the only finite solution of Schrodinger equation is ψ(x) = 0, signifying that the particle is never found in the negative x region. In the region (0 < x < a), the potential energy is V(x) = -V_0 and Schrodinger equation has the form

The general solution of this second-order differential equation has the form
&ps;(x) = C sin (k_0x + γ),
where C and g are arbitrary constants. To ensure continuity of c(x) at x = 0, we shall set the constant g to zero to give
ψ(x) = C sin k_0x --2
In the region (a < x < &infinity;), the potential energy is zero and schrodinger equation has the form

The general solution is

where A and A_0 are arbitrary constants. To ensure that the eigenfunction is finite at infinity, we set A_0 to zero to give a solution which falls off exponentially with x:
---3
Our next task is to join the solution given by Eq. (2), which is valid in the region (0 < x < a), onto the solution given by Eq. (3), which is valid in the region (a < x < &infinit;). As mentioned earlier, we shall require the eigenfunction and its first derivative to be continuous at x = a. Continuity of ψ(x) gives
---4
and continuity of dψ(t)/dx gives
---5
If we divide Eq. (5) by Eq. (5), we obtain
---6
Equation (6) sets the condition for a smooth join at x = a of the functions C sin k0x and Aeÿax. It is a non-trivial condition which is only satisfied when the parameters k0 and a take on special values. And once we have found these special values, we will be able to find the binding energies of the bound states
from ε = h^2a^2/2m.
To find these binding energies, we note that a and k0 are not independent parameters. They are defined by
---7
which imply that
---8
Thus, we have two simultaneous equations for a and k0, Eq. (6) and Eq. (8). These equations may be solved graphically by finding the points of intersection of the curves
---9
as illustrated in Figure below.
Inspection of Figure shows the number of points of intersection, and hence the number of bound states, increase as the well becomes deeper. In particular, there are no bound states for a shallow well with
---10
Graphical solution of the simultaneous equations a = k_0cot k_0a and a^2 + k0^2 = w^2. The units of k0 and a are p=a. Three values for the well-depth parameter, w = π/a, w = 2π/a and w = 3π/a, are labelled by (A), (B) and (C), respectively. For (A) there is one point of intersection and one bound state, for (B) there are two points of intersection and two bound states and for (C) there are three points of intersection and three bound states.
w < π/2a ;
There is one bound state when

and two bound states when

and so on. We can then find the binding energies of these bound states.

## Paradox of highest ordirnal number

ordinal number is defined as a class of series. It is is the class of generating relations of series. A series can be of many type. The paradox is :
Let us arrange all the ordinals , in orders of magnitude, from 0 up to N. But in that case the total ordinal numbers will be 1+ N which is larger than N. Hence there is no highest ordinal number. You can not escape this dilemma by saying that it is the series has a last term. But in that case , equally the series has some other odinal greater than any term of the series. Hence the paradox is inescapable.
The arithmetic of ordinal and cardinal numbers are very interesting and well defined by cantor. Somebody has famously qouted that Cantor has created a paradise from which no one can expell us. Cardinal number of a set is the class of all simillar classes. For more detailed analysis follow this page.

## Rydberg's states

Rydberg's state is also known as Rydberg's series. It is given by the formula

Schrodinger equation in cartesian coordinates

Solution of wavefunction of nitrogen aton is

The inception of quantum mechanics dates back to the blackbody radiation. It was long been known that a hot body emits radiation which varies r/>with the temparature of the body. But the accurate law of that radiation was unkown until Max planck showed that the radiation happens in discrete packet known as quanta. The theory of blackbody radiation is somewhat complicated and lengthy topic. A few laws can be mentioned for brief analysis.

## Shah equation

When the temperature rises high enough that the typical kinetic energy of atoms reaches the energy required to ionize a hydrogen atom, then the material becomes ionized." In other words,

Hold on a moment -- that calculation indicates that stars must be REALLY hot in order for a collision to ionize a hydrogen atom in its ground state. But what about atoms in excited states? The first excited level, n=2, is more than halfway from the ground state to the ionization energy. If we include atoms of all levels in the calculations, we might uncover that even stars with relatively low temperatures can ionize most of the hydrogen in their photospheres.
In order to do this calculation properly, we need to form a weighted average of the fraction of neutral atoms in each energy state.

If we can form the partition function of all hydrogen atoms in the neutral phase -- call that ZI -- and the partition function of all hydrogen atoms in the ionized stage -- call that ZII -- then we can use the Saha equation to calculate the relative number of atoms in each ionization stage.

In this equation, ne is the number density of free electrons, and χ is the energy required to ionize a hydrogen atom from its ground state. Some useful diagrams and equations

Quantum simulation of atomic heat transport equation is

Electron revolves around nucleus and that makes them possess orbital angular momentum. The magnitude of this orbital momentum is quantized.

The angular momentum is a vector quantity, hence its direction must be specified to describe it completely. To specify the orientation or direction of an orbit, a reference is required. The direction of the magnetic field applied to the atom is chosen as the reference line. This line is along the z-axis. The rotating electron about the nucleus forms a current loop which has a magnetic moment µ = IA.
The energy of loop-field system is given by U = -µ. B = – |µ| |B| cosθ, where θ is the angle between the magnetic moment μ and magnetic field B. As such classically any energy value between -µB to +µB is possible for the loop. An electron orbiting around the nucleus in an atom possess angular momentum L which interacts with external applied magnetic field B.
The phenomenon of quantization of L in the direction of magnetic field B is commonly known as space quantization.

The direction of B points towards the Z axis. As such the component of L along Z-direction is given as LZ = ml, h, the ml can table the values from –l to +l including zero.
Let us calculate the allowed projections of L for l = 2. The L can be visualized as a vector lying on the surface of a cone (see Figure below).

## Quantum mechanics formulas and matrix mechanics

creation and annihilation operators

Hamiltonian operators

Quantum field oiperators in terms of annihilation and creation operators

anti-commutators are defined using second brackets as

More formulas of creation and annihilation operators

## Quantum cosmology

The Wheeler-DeWitt equation which is a zero-energy Schrödinger-like equation for the wave function of the Universe. conjugate momenta is

Putting the above equations together we obtain

## Time in Canonical Quantum Gravity

the wave function of the universe Ψ depends only on the spacelike three-geometry (this gives origin to the “problem of time” mentioned in the introduction). This is due to the diffeomorphism invariance of general relativity (or equivalently, its general covariance), which implies that a particular (external) time coordinate should not be present in the equations (see Hamber).

Consequence is that

Semiclassical Approximation and the WdW equation

Where S₀ is a solution of the Hamilton-Jacobi equation for the classical theory, namely

Proceeding like in the standard quantum mechanics, and a few simple steps (see Hamber), one obtains:

where the ϕ superscript indicates that matter fields were included and the following definition of time was used:

## Semi empirical mass formula

Semi empirical mass formula can be used to calculate binding energy of an atom. This is mathematically expressed with

Balmer series is a series which represents atomic spectrum .

More lectures on quantum mechanics

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